One parameter deformation of the gamma function and some associated inequalities

Objective: In this study, the definition, properties, and relationships between the gamma function and a one parameter deformation of the gamma function (v-generalization, v-analog), called the v-gamma function are investigated. The aim is to provide some inequalities that include the v-analog of the gamma function. Materials and Methods: All articles and books on the subject are the research materials. The articles given in the references section were examined as a research technique and then the study was carried out using mathematical proof methods. Results:This thesis consists of four chapters. Following the introduction, the second chapter provides the basic definitions and theorems that will be used in other chapters of the thesis. In the third chapter, the definition of a one parameter deformation of the gamma function (v-generalization, v-analog) is given and some of its properties were examined. Certain inequalities involving the v-gamma function are expressed using Hölder and Minkowski inequalities. The fourth chapter present an asymptotic expansion of the v-gamma function. Additionally, the definition of the v-beta function, a one parameter deformation of the beta function is provided, and the limit expressions for the v-gamma and v-beta functions are demonstrated. Finally, by leveraging various properties of the v-digamma and v-polygamma functions, some inequalities involving the v-gamma, v-digamma, and v-polygamma functions are presented. Conclusion: In this master's thesis the v-gamma function which is a one parameter deformation of the gamma function and in association with that v-beta, v-digamma and v-polygamma functions have been introduced, and their properties have been thoroughly examined. Furthermore, results and inequalities involving the v-gamma, v-digamma and v-polygamma functions have been optained.